DT. N°. 2018-004

Serie de Documentos de Trabajo Working Paper series

Abril 2018

Los puntos de vista expresados en este documento de trabajo corresponden a los de los autores y no reflejan necesariamente la posición del Banco Central de Reserva del Perú.

The views expressed in this paper are those of the authors and do not reflect necessarily the position of

the Central Reserve Bank of Peru

BANCO CENTRAL DE RESERVA DEL PERÚ

Measuring the Stance of Monetary Policy in a Time-Varying World

Fernando Pérez Forero*

* Banco Central de Reserva del Perú

Measuring the Stance of Monetary Policy in a Time-Varying

World ∗

Fernando J. Perez Forero†

5th April 2018

Abstract

Knowing the stance of monetary policy is a general theme of interest for academics, policymakers and private sector agents. The mentioned stance is not necessarily observable, sincethe Fed have used different monetary instruments at different points in time. This paperprovides a measure of this stance for the last forty five years, which is a weighted average ofa pool of instruments. We extend Bernanke and Mihov (1998)’s Interbank Market model byallowing structural parameters and shock variances to change over time. In particular, wefollow the recent work of Canova and Perez Forero (2015) for estimating non-recursive TVC-VARs with Bayesian Methods. The estimated stance measure describes how tight/loose wasmonetary policy over time and takes into account the uncertainty related with posteriorestimates of time varying parameters. Finally, we present how has monetary transmissionmechanism changed over time, focusing our attention in the period after the Great Recession.

C11, E51, E52, E58

SVARs, Interbank Market, Operating Procedures, Monetary Policy Stance, Time-varyingparameters, Bayesian Methods, Multi-move Metropolis within Gibbs Sampling

∗This paper is a modified version of the chapter 2 of my dissertation titled Essays in Structural Macroe-conometrics. I would like to thank Fabio Canova and Kristoffer Nimark for their advising during my PhD atUniversitat Pompeu Fabra. I would also like to thank Christian Matthes, Luca Gambetti, Marek Jarocinski,seminar participants of Universitat Pompeu Fabra (2011-2013), seminar participants at the Central ReserveBank of Peru, CEF 2017 participants at New York and LACEA-LAMES 2017 participants at Buenos Aires andMarco Vega for their helpful comments and suggestions. The views expressed are those of the author and do notnecessarily reflect those of the Central Bank of Peru. All remaining errors are mine.†Head of Monetary Programming Department, Central Reserve Bank of Peru (BCRP), Jr. Santa Rosa 441,

Lima 1, Peru; Email address: fernando.perez@bcrp.gob.pe

1

1 Introduction

Knowing the stance of monetary policy is a general theme of interest for academics, policy

makers and private sector agents. It provides an important piece of information to understand

the current state of the economy and contributes to the expectations formation of future states.

Despite its importance, it has been difficult to have an exact measure of this stance in the past,

given the lack of consensus on which were the instruments of monetary policy and operating

procedures at each point in time. In the recent years this task has turned even more difficult

because of the introduction of the so-called Unconventional Monetary Policies (UMP) and be-

cause the Federal Funds Rate (FFR) hit the Zero-Lower-Bound (ZLB), given that the latter

used to be considered as the core instrument at least for the last two decades. The purpose

of this paper is to provide a measure of the policy stance that takes into account changes over

time of the operating procedures of the Fed.

In practice, Monetary Policy is implemented through the intervention of the Fed in the

interbank market of reserves. In this market, each participant has to meet a reserve requirement

set by the Fed in advance. To do that, the banks use the interbank loans market. Therefore,

the banks that have a deficit in reserves can borrow funds from the ones that have excessive

reserves. These loans are granted only if the borrowers have an amount of collateral equivalent

to the amount of credit that is going to be lent from another bank. The equilibrium price

of this market is the interbank interest rate, i.e. the FFR. In this context, the Fed performs

open market operations (OMO) in order to set the supply of reserves and a thereby affect

the equilibrium outcome of this market. In particular, if a bank can not meet its reserve

requirement, it will need to borrow reserves from the Fed at the Discount Window (DW), and

the obtained funds will be called Borrowed Reserves (BR). Given the Total Reserves (TR) stock

that market participants have at the end of the period, the difference is called Non-Borrowed

Reserves (NBR=TR-BR), i.e. Reserves that are obtained through open market operations.

Finally, the way these operations (OMO and DW) are implemented is what we call the Fed’s

operating procedures. In particular, if we take a look to the recent monetary history of the

United States, we can find evidence of different episodes with different operating procedures.

2

The implementation in each case depends on what was the Fed’s target at each point in time,

e. g. borrowed reserves, non-borrowed reserves, total reserves or the federal funds rate (see,

Cosimano and Jansen (1988), Cosimano and Sheehan (1994), Bernanke and Mihov (1998),

among others).

The empirical approach in this paper is based on Structural Vector Autorregressions (SVARs).

These time series models have been popular for identifying and assessing the dynamic effects

of monetary policy shocks. When using these models, the FFR has been considered as the core

instrument since the seminal work of Bernanke and Blinder (1992)12. On the other hand, Chris-

tiano and Eichenbaum (1992) and Strongin (1995) used reserves of banks as a monetary policy

instrument. Bernanke and Mihov (1998) reconcile these two strands of the literature with an

eclectic approach that identifies monetary policy shocks as a linear combination of innovations

in different instruments and, subsequently Christiano et al. (1999) summarizes this empirical

literature and takes stock of what we learned about SVARs and monetary policy shocks during

the 1990’s decade. In the previous decade (2001-2010) the SVARs have been extended to solve

some problems related with the identification procedures. First, Bernanke et al. (2005) sug-

gest that a large amount of information needs to be used to solve the price-puzzle. In second

place, the instability of parameters has been taken into account and also regime changes were

explicitly discussed (see Cogley and Sargent (2005), Primiceri (2005) and Sims and Zha (2006)).

Although useful, all these extensions still considered the FFR as the core instrument for mone-

tary policy. Clearly, the approaches that used the FFR as policy instrument are unsuitable to

discuss the role of the UMPs and the interest rate at the ZLB. In this regard, recent attempts

to identify monetary policy with the interest rate at the ZLB and the active use of the UMPs

can be found in Baumeister and Benati (2013) and also Peersman (2011). In particular, these

authors use sign restrictions as in Canova and De Nicolo (2002) and Uhlig (2005), and they

identify the UMP shocks that are different from standard FFR innovations. UMP has different

dimensions, and an innovation in each of them must be associated with monetary policy actions.

The problem is that it is difficult to interpret the role of the shocks for the sample portion before

1See also Sims (1986) and Leeper et al. (1996).2See Rubio-Ramırez et al. (2010) and Kilian (2012) for recent surveys related with SVARs and identification.

3

the financial crisis. As a result, considering a different policy shock for each UMP dimension

(as in Baumeister and Benati (2013) and Peersman (2011)) is not necessarily a good strategy.

We believe that the strategy of identifying monetary policy shocks as a linear combination of

innovations in different instruments, as in Bernanke and Mihov (1998), should be reconsidered,

given the various dimensions of UMPs (e.g. to influence Financial Markets conditions through

Large Scaled Asset Purchases, Forward Guidance, Direct Financial Intermediation, Quantita-

tive Easing) together with the FFR3. Thus, in this paper we use Bernanke and Mihov (1998)’s

approach to take into account the multi-dimensionality of monetary policy and the possibility

that different instruments matter at different points in time.

The building blocks of our approach are as follows. Bernanke and Mihov (1998) characterize

Federal Reserve’s operating procedures and provide a measure of the stance of monetary policy

for the period 1965-1996. Essentially, their model has an interbank market of reserves where

monetary policy can be implemented, depending on the parameter values of the model, by

targeting either interest rates (price of reserves) or the supply of reserves. What makes this

model useful is its capability to identify monetary policy shocks for different contexts and

instruments (see eq. (12) in the mentioned reference). These authors make explicit their concern

about stability of parameters along their sample of analysis because the operating procedures

might have changed over time (e.g. Volcker’s experiment in early 1980s or the recent UMPs

in our case). Instead, when measuring policy stance one should take into account that the

weight of each instrument is likely to be time varying. These weights are nonlinear functions

of the estimated structural parameters. Thus, in order to study the posterior distribution of

the path of the monetary policy stance and the weights taken by each component, we follow

Canova and Perez Forero (2015), who extend the setup of Time-Varying Coefficients (TVC)

VARs with Stochastic Volatility (Primiceri, 2005) to deal with non-recursive and potentially

over-identified SVAR models. In particular, we extend the framework used by Bernanke and

Mihov (1998) to account for the monetary policy (UMPs) practices. The model can be used

to study the role of Quantitative Easing (QE), since it identifies demand and supply shocks to

3See Williams (2011), Williams (2012b) and Williams (2012a) for a detailed description of UMP. See alsoBorio and Disyatat (2010) and Cecioni et al. (2011) for a thorough survey of the different dimensions of UMPs.

4

reserves and the discount window operations shocks. However, the model needs to be slightly

modified in order to capture Large Scaled Asset Purchases (LSAPs) and Forward Guidance (the

announcement of future path for interest rates), actions aimed to affect medium and long-term

interest rates. Thus, we include the shadow interest rate as a proxy of these measures (Wu and

Xia, 2015). It is important to remark that this is different from including a specific equation

for UMP shocks4. One potential limitation of our approach is that we do not make explicit the

role of communication (i.e. FOMC meeting Releases, Minutes, Speeches, etc.) and the recently

introduced interest paid for holding reserves. Our strategy to characterize the Monetary Policy

Stance is robust in terms of specification, since we are allowing structural parameters in both

policy and non-policy blocks to vary over time. On the other hand, we believe that part of the

effect of FOMC communication is captured through changes in shadow interest rate, which is

included in our SVAR.

We find that the stance of monetary policy has varied quite a lot over the last 45 years. It

was loose for the first half of the 1970s and roughly neutral for the second half, it becomes tighter

at the beginning of Volcker’s period, i.e. the so-called Volcker’s disinflation experiment (1980-

1982) and then becomes loose again. Volcker’s period ends with a relatively tight stance but

showing more uncertainty than before. Greenspan’s first ten years (1987-1996) exhibit a tight

stance with a short period of loose policy in 1989. A long episode of loose stance (1996-2001)

is observed with a subsequent neutral stage (2002-2003). Last Greenspan’s years (2003-2005)

display a relatively tight stance but shows an upward trend starting in late 2004. The stance

turns to be loose when Bernanke’s period starts until the outbreak of the Great Recession in

2007:Q4, when the stance turns to be tight again since 2008:Q4. We finally observe a reversal

of this pattern after the implementation of UMPs, when the stance turns to be relatively loose

in 2011-2013. This result is also in line with Beckworth (2011), who claims that the Monetary

Policy Stance was relatively tight in 2008. After the tapering talk of may 2013, the stance

turned to be less loose until our days, a result that is in line with the recent events observed,

i.e. the FFR was raised in December 2015, and it is expected to be hiked again in December

4Reis (2009), Blinder (2010), Lenza et al. (2010) and Hamilton and Wu (2012) present the main characteristicsof UMPs, emphasizing the role of yield curve spreads as a powerful indicator that summarizes both credit policyas well as the expectations of future paths for interest rates (Forward Guidance).

5

2016. The relative weights of these instruments are time-varying, where the most important

result is the weight of zero for the FFR at the end of the sample, consistent with the binding

ZLB. What matters here is the fact that the model is capable of capturing significant changes

in operating procedures.

Model estimates allow us to explore time variations in the transmission of policy shocks.

Overall, the transmission of monetary policy shocks is stable for a large portion of our sample,

but it exhibits significant changes after the outbreak of the Great Financial Crisis and the

achievement of the ZLB. We find that the effect of expansionary policy shocks on the spreads

is positive before 2007, but turns to be negative afterward. The latter is consistent with the

purpose of UMPs, i.e. since the FFR is constant, the objective is to cut medium and long term

interest rates. We also find evidence of a vanishing liquidity effect over time and that monetary

policy shocks became relatively more volatile during Volcker’s episode and during the Great

Recession.

In sum, the approach this paper presents is capable of capturing changes in monetary policy

implementation across different episodes. We present a monetary policy stance index that might

be be useful for both policy makers and researchers. However, more work is needed for exploring

the explicit role of communication in UMPs, the announcement of future paths of interest rates

and credibility. We believe that these type of issues should be explored in a richer setup and

therefore we leave it for future agenda. In this regard, some structural models that incorporate

different dimensions of UMPs can be found in Gertler and Karadi (2011), Curdia and Woodford

(2011) and Chen et al. (2012).

The document is organized as follows: section 2 presents the Structural VAR model used for

the analysis, section 3 describes the estimation procedure, section 4 presents an estimate of the

monetary policy stance, sections 5 and 6 explore the transmission mechanism and the volatility

of monetary policy shocks, respectively and section 7 concludes.

6

2 The Model

2.1 A Structural Dynamic System

We are interested in characterize a dynamic setting in order to identify monetary policy shocks

and for that purpose we closely follow the method proposed by Bernanke and Blinder (1992)

and Bernanke and Mihov (1998). That is, assume that the structure of the economy is linear

and given by

Yt = cnpt Dt +

p∑i=0

Ri,tYt−i +

p∑i=0

Ti,tPt−i + Cnpt vnpt

Pt = cptDt +

p∑i=0

Si,tYt−i +

p∑i=0

Gi,tPt−i + Cpt vpt

where Yt is a vector of macroeconomic variables, Pt is a vector of monetary policy instruments,

cnpt and cpt are matrices of coefficients on a vector of deterministic variables Dt and vnpt and vpt

are vectors of structural shocks that can hit the economy at any point in time t = 1, . . . , T with

vkt ∼ N(0,Σk

tΣk′t

); k = np, p

where ΣktΣ

k′t is a diagonal positive definite matrix and Cov (vnpt , vpt ) = 0.

Notice that here we allow for potential time variation in matrix coefficients and variances

and therefore we include the index t for each of them. As in the mentioned references, we

assume that macroeconomic variables Yt do not react within the next period to innovations in

policy instruments, i.e. T0,t = 0 ∀t, so that

Yt = cnpt Dt +

p∑i=0

Ri,tYt−i +

p∑i=1

Ti,tPt−i + Cnpt vnpt (1)

Pt = cptDt +

p∑i=0

Si,tYt−i +

p∑i=0

Gi,tPt−i + Cpt vpt

where we assume that a period t is a quarter. For now we can say that the structure of the

economy (1) takes the form of a system of Vector Autorregressions (VAR) of order p. In order

7

to be in line with a textbook representation of this type of system we present a more compact

setup. Denote the vector of variables yt ≡ [Y ′t , P′t ]′, the vector of intercepts ct ≡

[cnp′t , cp′t

]′and

the matrices

At ≡

A11,t A12,t

A21,t A22,t

=

I −R0,t 0

−S0,t I −G0,t

(2)

Ai,t ≡

Ri,t Ti,t

Si,t Gi,t

; i = 1, . . . , p

Ct ≡

C11,t C12,t

C21,t C22,t

=

Cnpt 0

0 Cpt

; Σt ≡

Σnpt 0

0 Σpt

so that the model can be re-expressed as a Structural VAR with time-varying coefficients:

Atyt = ctDt +A1,tyt−1 + . . .+Ap,tyt−p + Ctvt (3)

However, the economic model expressed in its structural form cannot be directly estimated

without additional assumptions. That is, in order to identify the vector of structural shocks[vnp′t , vp′t

]′we should ask which form take matrices At and Ct, which are governed by structural

parameters. We will describe the structural model in subsection 2.3 but first we describe the

setup in detail in next subsection.

2.2 Basic setup

Consider a vector of M variables yt(M×1) with data available for T periods. I assume that

the data generating process for yt is the reduced-form version of the model (3), i.e. a VAR(p)

process such that:

yt = B0,tDt +B1,tyt−1 + ...+Bp,tyt−p + ut; t = 1, . . . , T (4)

where B0,t ≡ A−1t ct is a matrix of coefficients on a M × 1 vector of deterministic variables

Dt and Bi,t ≡ A−1t Ai,t; i = 1, . . . , p are M × M matrices containing the coefficients on the

8

lags of the endogenous variables and the error term is distributed as ut(M×1) ∼ N (0,Ωt), where

Ωt(M×M) is a symmetric, positive definite, full rank matrix for every t. Equation (4) is a reduced

form and the error terms ut do not have an economic interpretation. Let the structural shocks

be εt ∼ N (0, IM ) and the mapping between these shocks and their reduced form counterpart is

ut = A−1t CtΣtεt (5)

where At(M×M), Ct(M×M) and Σt(M×M) are defined in (2). In order to be in line with the

notation of previous subsection, we should note that εt ≡ Σ−1t vt is the normalized version of the

structural shocks. I substitute (5) into (4) so that we get structural form of the VAR(p) model:

yt = X ′tBt +A−1t CtΣtεt (6)

The matrix of regressors is X ′t = IM ⊗[D′t,y

′t−1, . . . ,y

′t−p]

is a M × K matrix where Dt

potentially includes a constant term, trends, seasonal dummy variables, etc and K = M ×

M + p ×M2. Parameter blocks(Bt, At, C

−1t , σt

)are treated as latent variables that evolve as

independent random walks:

Bt = Bt−1 + υt (7)

αt = αt−1 + ζt (8)

ct = ct−1 + %t (9)

log (σt) = log (σt−1) + ηt (10)

where Bt(K×1) =[vec (B0,t)

′ , vec (B1,t)′ , . . . , vec (Bp,t)

′]′ is a K×1 vector; αt and ct denote free

parameters of matrices At and C−1t , respectively5. In addition, σt(M×1) is the main diagonal of

5The reason of why we are focused on C−1t instead of Ct is for computational convenience after the construction

of the State-Space model (see Appendix A.4 for details). For instance, denote ct as the vector of free parametersof Ct. Then, if Ct is lower-triangular with ones in the main diagonal (which is indeed the case here), then theset of free parameters of C−1

t will be simply the vector ct = −ct. Thus, recovering the original parameters willbe straightforward.

9

Σt. Finally, the covariance matrix for the error vector is:

V = V ar

εt

υt

ζt

%t

ηt

=

IM 0 0 0 0

0 Q 0 0 0

0 0 Sa 0 0

0 0 0 Sc 0

0 0 0 0 W

(11)

The model presented captures time variations of different parameter blocks: i) lag structure

(7), ii) structural parameters (8) and (9) and iii) structural variances (10). In other words, the

model is capable of capturing the sources of potential structural changes, i.e. drifting coefficients

(Bt, αt, ct) or stochastic volatility (σt) without imposing prior information about specific dates

or number of structural breaks. In particular, in the process of identifying parameters that

affect the policy stance and represent the operating procedures, a subset of (αt, ct) will have a

major relevancy.

2.3 A Structural VAR model with an Interbank Market

Bernanke and Mihov (1998) presented a semi-structural VAR model that characterizes Federal

Reserve’s operating procedures. The purpose of this section is to present an extension of the

mentioned model in order to take into account different dimensions of UMP together with

conventional policy.

Consider the vector of variables yt = [xt, πt,∆Pcomt, TRt, FFRt, NBRt]′, where xt rep-

resents output growth rate, πt represents the inflation rate, ∆Pcomt is the growth rate of an

index of commodity prices, TRt is the total amount of reserves that banks hold at the Central

Bank, FFRt is the Federal Funds Rate in annual terms6 and NBRt is the total amount of non-

borrowed reserves (portion of reserves different from borrowed reserves at the discount window,

6We use the shadow interest rate estimated by Wu and Xia (2015) for the period 2008-2015.

10

i.e. NBR = TR−BR). Regarding the model specification, we re-write equation (5) as follows7

Atut = Ctvt

where we need to recall that vt = Σtεt is the re-scaled vector of structural shocks and ut

is the vector of reduced-form innovations. Moreover, recall the system partition described in

subsection 2.1. That is, there is a non-policy block and a policy block:

A11,t A12,t

A21,t A22,t

unpt

upt

=

C11,t C12,t

C21,t C22,t

vnpt

vpt

(12)

Within the non-policy block we can find output growth, inflation, and commodity prices growth

that is, unpt = [uxt , uπt , u

ct ]′ and vnpt = [vxt , v

πt , v

ct ]′. the system has M = 6 variables, where we

denote the number of non-policy variables as Mnp = dimunpt = 3. The policy block contains

the remaining variables of the system, i.e. Mp = M −Mnp = dimupt = 3, and we will call them

policy instruments. Here below we specify a sub-system of equations for the portion of upt that

is orthogonal to the non-policy block unpt . The set of assumptions embedded in the system of

equations above can be summarized as follows:

1. Non-policy block : First, recall that non-policy variables only react to policy changes with

some delay, i.e. according to (2), we have A12,t = 0(Mp×Mnp), ∀t. The intuition behind

this assumption is that the private sector considers lagged stance of policy as state vari-

able. That is, output growth and inflation will not change in the same quarter after an

innovation in a particular instrument that belongs to upt . Moreover, following Bernanke

and Mihov (1998), I will keep this non-policy block unmodeled and I will just assume that

7According to Amisano and Giannini (1997) and Lutkepohl (2005) (ch. 9), the model presented in is oneversion of the AB model. See the mentioned references for details.

11

A11,t is lower triangular, so that

A11,t =

1 0 0

απx,t 1 0

αcx,t αcπ,t 1

The ordering in this block is an open question and that is why the model is called semi-

structural8. Moreover, I also assume that innovations in unpt will affect the policy block

in the same quarter, i.e. A21,t is an unrestricted Mnp×Mp matrix of potentially non-zero

parameters (see Appendix A.4). In addition, according to (2), we have C12,t = 0(Mp×Mnp)

and C21,t = 0(Mnp×Mp), ∀t. We also assume that C11,t = IMp is the identity matrix,

which means that structural shocks vxt , vπt and vct only affect output growth, inflation and

commodity growth on impact, i.e. there are no cross-effects on impact.

Turning to the policy block, the next three equations have the aim to describe the In-

terbank Market of Reserves. That is, each period t banks have to meet their reserve

requirements determined by the Fed. The sum of the level of reserves across banks de-

termines the term ”Total Reserves” denoted by TRt. Moreover, these reserves pay an

interest that is closely related to the Federal Funds Rate, FFRt and a result the latter

is a relevant indicator for the demand of reserves. In order to meet their reserve require-

ments banks have two alternatives: they could get liquidity from the Discount Window

(”Borrowed Reserves”, BRt) or through Open Market Operations (”Non-Borrowed Re-

serves”, NBRt). Banks have a pool of assets that are used as a collateral in order to

get liquidity through the mentioned alternatives. To close the model, we assume that the

Federal Reserve regulates liquidity by deciding the amount of reserves that is going to be

injected through Open Market Operations. I will now proceed to describe the structural

equations that conform the Interbank Market. In line with equation (12), I express the

model in terms of reduced-form and structural innovations:

• Demand for Reserves equation: It represents the total demand for reserves of banks.

8However, Cogley and Sargent (2005) as well as Primiceri (2005) find that changing the ordering of thesevariables did not affect their results significantly.

12

In particular, the portion of uTRt which is orthogonal to the non-policy block depends

negatively on the Federal Funds Rate’s innovation uFFRt and vdt represents a shock

to reserves’ demand.

uTRt = −αd1,tuFFRt + vdt (13)

• Demand for discount window operations equation: Borrowed Reserves (BR) is the

portion of reserves obtained through the discount window. They depend positively

on the Federal Funds Rate’s innovations uFFRt and vbt represents a shock to the

discount window operations’ demand, a potential source of fluctuation that could

become relevant in episodes of financial stress or under BR targeting.

uBRt = uTRt − uNBRt = αb1,tuFFRt + vbt (14)

• Federal Reserve equation: It represents the money supply process, i.e. providing

enough liquidity through open market operations in order clear the money market.

The portion of uNBRt which is orthogonal to the non-policy block responds contem-

poraneously to shocks in spreads, demand for total and borrowed reserves. Every

other action unrelated with the mentioned shocks is called an exogenous monetary

policy shock, vst .

uNBRt = φdt vdt + φbtv

bt + vst (15)

Equations (13) and (14) and (15) also appear in the benchmark version from Bernanke and

Mihov (1998) and our contribution is the addition of time variation in structural parameters

plus the inclusion of the shadow interest rates. From the latter reference we import the idea

that each of the three variables is considered as a monetary policy instrument controlled by

the Fed. In this framework we abstract for the role of Fed’s monetary policy communication

(i.e. FOMC meeting Releases, Minutes, Speeches, etc.) as well as for interest paid by reserves.

In this regard, it is worth to say that our strategy is safe in terms of specification, since we

are allowing structural parameters from both policy and non-policy blocks as well as structural

variances to vary over time, i.e. since the model is an approximation that could be potentially

13

misspecified (indeed, the model is linear and not micro-founded), it is likely that posterior

estimates of structural parameters will vary across sub-samples. Therefore, it is even better to

allow for continuous drifting parameters9.

Recall (12) and consider now the sub-system of equations for the portion of upt that is

orthogonal to the non-policy block unpt , i.e. A22,tupt = C22,tv

pt . The system is10:

1 αd1,t 0

1 −αb1,t −1

0 0 1

︸ ︷︷ ︸

A22,t

×

uTRt

uFFRt

uNBRt

=

1 0 0

0 1 0

φdt φbt 1

︸ ︷︷ ︸

C22,t

×

vdt

vbt

vst

The latter system can be solved for structural shocks, i.e. vpt = C−122,tA22,tupt . In particular, the

bottom equation of this system corresponds to the monetary policy shock vst , that is:

vst = −(φbt + φdt

)uTRt +

(φbtα

b1,t − φdtαd1,t

)uFFRt +

(φbt + 1

)uNBRt (16)

The intuition behind equation (16) is that monetary policy actions can be expressed as a

linear combination of innovations of different instruments. That is, policy actions are not

only characterized as interest rate innovations as it is commonly suggested in the literature.

Therefore, (16) can be used to evaluate the Monetary Policy Stance.

Because we are allowing coefficients to vary over time, the weight of each instrument will turn

to be time-varying as well. These weights are nonlinear functions of structural parameters that

come from the estimated SVAR model, as it is explicit in (16). As a result, this characterization

of policy actions is robust to changes in the operating procedures during the sample of analysis.

For instance, consider the case of monetary policy conducted by interest rate setting. If the

ZLB is binding, then the FFR will no longer be the policy instrument, at least temporarily. As a

result, the Fed will re-design its operating procedures putting more weight on other instruments

9See Cogley and Yagihashi (2010), Chang et al. (2010) and Canova and Perez Forero (2015) for more detailsabout this issue.

10We have considered that now√var

(vbt)

= σbt/∣∣αb

1,t

∣∣. See also ?) for a similar approach using Bernanke and

Mihov (1998)’s model.

14

and assigning zero weight to the FFR11.

The sub-system has Mp = 3 variables. Therefore the variance covariance matrix of the

reduced form error terms upt will have 3 × (3 + 1) /2 = 6 parameters. On the other hand, the

vector of structural parameters has 7 elements, i.e.

θt =(αd1,t, α

b1,t, φ

dt , φ

bt , σ

dt , σ

bt , σ

st

)′These parameters are actually latent variables, however we set identifying restrictions to be in

line with the SVAR literature (see Canova and Perez Forero (2015) for details). Thus, to achieve

identification it is necessary to impose an additional restriction. Following Bernanke and Mihov

(1998), we can focus our attention on equation (16) and set restrictions such that monetary

policy is associated with a particular instrument, i.e. set restrictions such that all the brackets

are equal to zero except the one associated with our instrument of interest. Nevertheless,

since our sample covers a period where the Fed had different chairmen plus UMP, we have less

incentives in restricting our attention to a particular instrument. Instead, we want to capture

changes in operating procedures, which means that all the brackets in (16) should be potentially

different from zero. For that reason we set

αd1,t = 0 (17)

This restriction is in line with Strongin (1995) and Bernanke and Mihov (1998), and it means

that the demand of Total Reserves is inelastic with respect to the Federal Funds Rate, though

we have to take into account that this argument was used in a context where the Federal Funds

Rate was strictly positive. Our system of equation will be exactly identified after imposing (17)

and, as mentioned above, it implies that we are still free to consider different instruments of

monetary policy at given time. We also test the sensitivity and plausibility of this identification

restrictions in section .

In general, the estimation of the path for the structural parameters and the consequent

11See for instance Reis (2009) and Blinder (2010).

15

computation of vst in (16) will give us a measure of the monetary policy stance that internalizes

changes in operating procedures, i.e. time varying weights will shed light on the relative impor-

tance of each of the four instruments at each point in time. On the other hand, the estimated

paths for the variances will shed light on the relative importance of each of the shocks.

3 Bayesian Estimation

The purpose of this section is to describe the procedure in which the parameters of the sta-

tistical model described in (6) are simulated through Markov Chain Monte Carlo methods. In

particular, we are interested in the posterior distribution of latent variables described in (7),

(8), (9) and (10). I will adopt a Bayesian perspective following Primiceri (2005)’s Multi-move

Gibbs Sampling procedure. Moreover, we will sample structural coefficients from (8) and (9)

introducing two Metropolis-type steps, as suggested by Canova and Perez Forero (2015).

3.1 Data description

The time series used for the experiment are in monthly frequency and were taken from the

FRED Database and from the Federal Reserve Board website12. From the second database I

took Total Reserves of aggregated depository institutions, Nonborrowed Reserves of aggregated

depository institutions. The sample horizon is 1959:M1 - 2016:M9 (683 obs.), i.e. it includes

the whole period of the Great Recession.

Industrial Production, Consumer Price Index and the Commodity Price Index were ex-

pressed in annual growth rates. Federal Funds Rate is expressed in annual terms, including the

period of the shadow rate. Besides, to induce stationarity, Total and Nonborrowed Reserves

were standardized using the mean and standard deviation of Non-Borrowed Reserves for a win-

dow of 48 months. Alternatively, Bernanke and Mihov (1998) divide Total and Non-Borrowed

reserves by the average of Total Reserves using a window of 36 months. However this approach

is not useful for inducing stationarity anymore, given the recent changes in reserves13.

12https://fred.stlouisfed.org, https://www.federalreserve.gov/releases/h3/current/,13We have also tried to regress Total and Non-Borrowed Reserves onto a constant and linear trend with breaks

in the third quarter of 2008. However, the obtained residuals are extremely volatile.

16

3.2 Priors and setup

Priors are shown in Table 1 and they are chosen to be conjugated. As a result the posterior

distribution will be Normal and Inverted-Wishart for each corresponding case. Moreover, block

BT ’s posterior distribution is truncated for stationary draws. That is, the associated companion

form of the VAR (4) is computed for each draw of BT and it is discarded if it does not satisfy the

stability condition for each period t = 1, . . . , T . The latter procedure is captured by the indicator

function IB (.). In addition, I calibrate the prior for initial states of structural parameters using

the first τ = 120 observations (1959:M1 - 1969:M12) as a training sample, i.e. we estimate(B,VB

)via OLS and (α′, c′, σ′)′ via Maximum Likelihood1415.

Moreover, I calibrate k2Q = 0.5× 10−4, k2W = 1× 10−4, following Primiceri (2005), but also I

calibrate k2Sa= k2Sc

= 1× 10−1 to allow for significant time variation. Finally, lag length is set

to p = 2 as in the latter references.

Table 1: Priors

B0∼N(B,VB

)Q ∼ IW

(k2Q ·VB, τ

)α0 ∼ N (α, Idimα) Sa∼ IW

(k2Sa· Idimα, 1 + dimα

)c0 ∼ N (−c, Idim c) Sc∼ IW

(k2Sc· Idim c, 1 + dim c

)log (σ0) ∼ N (log (σ) , IM ) Wi∼ IG

(k2W , 1/2

), i = 1, . . . ,M

Since it is assumed that blocks(BT , αT , cT , σT

)follow random walks according to equations

(7), (8), (9) and (10), we use the mean and the variance of the priors of B0, α0, c0, log (σ0) to

initialize the Kalman Filter at each iteration. The sampling procedure is described in the next

subsection.

14The MATLAB code csminwel.m from professor C. Sims website is used(http://sims.princeton.edu/yftp/optimize/mfiles/). I have chosen randomly 100 different starting pointsin order to find a global maximum.

15Alternatively, we could use a Minnesota-style prior for calibrating(B,VB

)(see Del Negro (2003), Canova

(2007) (ch.10), among others), but we do not cover this issue here.

17

3.3 Sampling parameter blocks

We have to sample parameter blocks(BT , αT , cT , σT , sT , V

)and we do it sequentially using

the logic of Gibbs Sampling (see Chib (2001)). The block sT is an auxiliary one used as an

intermediate step for sampling σT , see Kim et al. (1998). The sampling algorithm is as follows:

1. Set an initial value for(BT

0 , αT0 , c

T0 , σ

T0 , s

T0 , V0

)and set i = 1.

2. Draw the reduced form coefficients BTi from p

(BTi | αTi−1, cTi−1, σTi−1, sTi−1, Vi−1

)· IB

(BTi

).

3. Draw the structural parameters αTi from p(αTi | BT

i , cTi−1, σ

Ti−1, s

Ti−1, Vi−1

).

4. Draw the structural parameters cTi from p(cTi | BT

i , αTi , σ

Ti−1, s

Ti−1, Vi−1

).

5. Draw volatilities σTi from p(σTi | BT

i , αTi , c

Ti , s

Ti−1, Vi−1

).

6. Draw the indicator sTi from p(sTi | BT

i , αTi , c

Ti , σ

Ti , Vi−1

).

7. Draw the hyper-parameters Vi from p(Vi | αTi , cTi , yTi , σTi , sTi

).

8. Set BTi , α

Ti , c

Ti , σ

Ti , s

Ti , Vi as the initial value for the next iteration. If i < N , set i = i+ 1

and go back to 2, otherwise stop.

The indicator function IB (.) truncates the posterior distribution of BT for draws that exhibit

a stationary companion form for t = 1, . . . , T . I perform N = 100, 000 draws discarding the first

50, 000 and I store one every 100 draws for the last 50, 000 to reduce serial correlation. Specific

details regarding the sampling algorithm are described in Canova and Perez Forero (2015) and

also in Appendix A.2 and B for the diagnosis of convergence.

4 The Stance of Monetary Policy

Using equation (16) and the identification restrictions (17); replacing the reduced form inno-

vations by the data included in the VAR; and renaming the resulting index as the monetary

policy stance (MPS), the expression of interest is:

MPSt = −(φbt + φdt

)uTRt +

(φbtα

b1,t

)uFFRt +

(φbt + 1

)uNBRt (18)

18

We compute the posterior distribution for MPSt at each point in time using posterior estimates

of the parameters.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-80

-60

-40

-20

0

20

40

60

80

1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 2012 2015

Greenspan Volcker Bernanke Burns / Miller

Yellen

Figure 1: Posterior distribution of the Monetary Policy Stance, median value and 90 percentconfidence bands

Figure 1 depicts the Monetary Policy Stance for the period 1974-2016. To the best of our

knowledge, this is the first time that the path of the policy stance index is produced with

error bands. The Stance of Monetary Policy is loose for a first half of the decade of 1970s,

a situation that is reverted in late 1970s. Moreover, we can see an even tighter stance at the

beginning of Volcker’s period, i.e. the so-called Volcker’s disinflation experiment (1980-1982)

19

with a subsequent loose stance. Then, Volcker’s period ends with a relatively tight stance .

Greenspan’s first five years (1987-1992) exhibit a tight stance. On the other hand, a large

episode of loose stance (1992-1999) is observed. Then we have a tight stance for a short period

(1999-2003). Last Greenspan’s years (2003-2005) exhibit a relatively loose stance. The stance

turns to be tight when Bernanke’s period starts until the outbreak of the Great Recession in

2007:Q4, when stance turns to be loose again. Part of the explanation of the reversion in

this pattern is the implementation of UMPs, when the stance turns to be relatively loose at

the end of our sample (2011-2016). This result is also in line with Beckworth (2011), who

claim that the Monetary Policy Stance was relatively tight in 2008. Moreover, although the

scales are different and besides the fact that we provide error bands for the estimated values, the

estimated index qualitatively replicates the periods of loose and tight monetary policy suggested

by Bernanke and Mihov (1998) and Boschen and Mills (1991). Therefore, the contribution of

our analysis is precisely the quantification of the uncertainty and the addition of the recent

period of unconventional monetary policies.

20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-80

-60

-40

-20

0

20

40

60

80

1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 2012 2015

Figure 2: Monetary Policy Stance and NBER recession dates (shaded areas)

It is important to associate the monetary policy stance with each appointed chairman. In

addition, we show the same stance but including the NBER recession dates in shaded areas

in Figure 2. The first aspect that needs to be mentioned is the one related with Volcker’s

disinflation period (1980-1982), where we can observe a sharp fall in the stance in late 1980

related with a sharp fall in long term interest rates16. In general, we observe a raise in our

MPS index during recessions, meaning that the usual reaction is to implement an expansionary

monetary policy in order to revert the negative state of the economy.

16See Goodfriend and King (2005).

21

5 The Transmission Mechanism of Monetary Policy revisited

In this section we explore the transmission of monetary policy shocks (εst ) on the interbank

market and the aggregate economy. As a consequence of the continuous time variation of

parameter values, it will be possible to trace impulse responses along the time dimension and

explore their evolution over time. Let the impulse response function be

∂yt+j∂εt

= Fj

(Bit+ji=t , At, C

−1t ,Σt

); j = 0, 1, . . . (19)

where Fj (.) depends on the companion form matrix of (4) for periods t, t+ 1, . . . , t+ j and the

blocks At, C−1t , Σt and depends on when the shock occurs. The exact form of equation (19) is

∂yt+j∂εt

= Et

[J(j−1k=0A

ct+j−k

)J′Ht (εt − εt)

](20)

where Ht = A−1t CtΣt and Act is the companion form matrix of (4). Details on the derivation of

equation (20) can be found in Canova and Ciccarelli (2009).

Figure 3 depicts the response of each variable after an expansionary policy shock (εst ) in 1996,

a date that we associate with normal times. First, output growth and inflation exhibits a hump-

shaped response where the peaks are 6 and 12 quarters after the shock. The expansionary policy

produces an increase in Total Reserves (TR) as well as a liquidity effect, i.e. Non-Borrowed

Reserves (NBR) and Federal Funds Rate (FFR) move in opposite directions after the shock

occurs.

22

3 6 9 12 15 18−0.5

0

0.5

1

1.5

2

2.5Output, 1996 m3

3 6 9 12 15 18−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4Inflation, 1996 m3

3 6 9 12 15 18−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2Pcom, 1996 m3

3 6 9 12 15 18−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0TR, 1996 m3

3 6 9 12 15 18−0.4

−0.2

0

0.2

0.4

0.6

0.8

1FFR, 1996 m3

3 6 9 12 15 18−0.5

0

0.5

1NBR, 1996 m3

Figure 3: Responses to Monetary Policy shocks in 1996, 90 percent confidence interval

Figure 4 depicts the response of each variable after an expansionary policy shock (εst ) in 2012,

a date that is associated with the aftermath of the last financial crisis. As it is noticeable, the

transmission mechanism of monetary policy has been altered. In particular, the response of

inflation is stronger and more persistent. As a matter of fact, the response of the FFR turns

to be small and almost insignificant, since the target rate is in an interval close to zero, i.e.

between 0 and 0.25 basis points.

23

3 6 9 12 15 18−0.5

0

0.5

1

1.5

2Output, 2012 m3

3 6 9 12 15 18−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Inflation, 2012 m3

3 6 9 12 15 180

0.5

1

1.5Pcom, 2012 m3

3 6 9 12 15 18−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0TR, 2012 m3

3 6 9 12 15 18−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7FFR, 2012 m3

3 6 9 12 15 18−0.5

0

0.5

1NBR, 2012 m3

Figure 4: Responses to Monetary Policy shocks in 2012, 90 percent confidence interval

Given this evidence of changes in the monetary transmission mechanism, we want to dig

into a particular issue of interest, named the Vanishing liquidity effect. According to the SVAR

literature on monetary policy shocks17, there is evidence of a temporary negative reaction of

interest rates that vanishes in the long run after an expansionary shock, i.e. an increase in money

supply. This pattern is manifested, in particular, when Non-Borrowed Reserves are included

in the SVAR model instead of money aggregates such as M1 or M2. As a result, our model is

capable of reproducing this feature and, furthermore, is capable of saying that this pattern has

been changing over time. Figure 5 depicts the evolution of the response of the Federal Funds

Rate, where it can be observed a strong response previous to late 1980s, and then a decreasing

response over time until hitting the Zero Lower Bound (ZLB) in 2009.

17See. e.g. Christiano and Eichenbaum (1991), Strongin (1995), Christiano et al. (1999), among others.

24

36

912

1518

19791984

19891994

19992004

20092014

−0.4

−0.2

0

0.2

0.4

Figure 5: Response of the Federal Funds Rate to an expansionary policy shock

Overall, there is evidence of time-varying monetary transmission mechanism; this insta-

bility can be associated with changes in monetary policy conduction with different operating

procedures. Note also that our result controls for changes in the private sector behavior, since

we allow the parameters of non-policy block to vary over time as well, and these changes are

correlated with changes in the policy design (see Canova and Perez Forero (2015)).

6 The Systematic and Non-systematic components of Monetary

Policy

The recent literature of Time varying coefficients (TVC), Stochastic Volatility and Monetary

Policy claims for more volatile policy shocks in early 1980s, a date that is associated with changes

in monetary policy conduction and the use of atypical instruments (see Primiceri (2005), Sims

and Zha (2006), Justiniano and Primiceri (2008), Canova et al. (2008), Canova and Gambetti

(2009) among others). However, we believe that part of this result is driven by the fact that

25

most of these models only allow for a single monetary policy instrument, i.e. the short term

interest rate. On the other hand, as we pointed out above, here we identify monetary policy

shocks including different instruments and controlling for changes in the systematic component.

For instance, recall the policy rule equation (15):

uNBRt = φdt vdt + φbtv

bt + vst

The systematic component is represented by the policy coefficients φt =(φdt , φ

bt

)′= −ct and the

non-systematic one is governed by the shock vst ∼ N(

0, (σst )2)

. We will call the evolution of φt

as changes in the systematic component and the evolution of σst as changes in the non-systematic

one.

Figure 6 depicts the evolution of the components of vector φt. In particular, we observe a

relatively unstable behavior of φdt starting in 2000 and magnified during the crisis episode. That

is, we clearly observe a change in the operating procedures during this era, which is in line with

the motivation of this paper, i.e. quantifying the implementation of Unconventional Monetary

Policies. In particular, the changing reaction function of the Fed reflects the fact that liquidity

demand shocks became more important during the crisis episode. On the other hand, regarding

the discount window shocks, our results reflect that they were significant during the 1970s and

early 1980s, and now the became less important for the Fed.

1980 1990 2000 2010

0.7

0.8

0.9

1

1.1φd

1980 1990 2000 2010−0.04

−0.03

−0.02

−0.01

0

0.01φb

Figure 6: Policy rule coefficients, median value and 90 percent c.i.

26

On the other hand, Figure 7 shows the evolution of the non-systematic component. In line

with the mentioned references, we observe a high level of volatility in 1970s and early 1980s, but

until 1985. On the other hand, we observe a second episode of high volatility starting in 2007,

i.e. an episode related with the Great Financial Crisis. This jump is probably capturing the

other dimensions of Unconventional Monetary Policy not related with Reserves and affecting

future interest rates, e.g. Direct Financial Intermediation as in Gertler and Karadi (2011). For

a thorough description of different dimensions of UMPs see also Reis (2009), Borio and Disyatat

(2010), Curdia and Woodford (2011), Cecioni et al. (2011) among others.

1975 1980 1985 1990 1995 2000 2005 2010 20150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Figure 7: Standard Deviation of Monetary Policy shocks

Finally, two things are worth to recall. In first place, the setup of the model is such that

the variances of structural shocks evolve independently, i.e. matrix W in (11) is set as diagonal.

Therefore, we cannot attribute the latter result to the fact that these shocks were correlated.

As a matter of fact, by definition these structural shocks are orthogonal to any other source of

disturbance. Second, we are capturing the portion of structural change that can be considered as

27

non-systematic, i.e. different than changes in the operating procedures represented by changes

in parameters of matrices At and Ct.

7 Concluding Remarks

We have estimated the monetary policy stance index for the U.S. economy for the period 1974-

2016, taking into account the time-varying operating procedures. To the best of our knowledge,

this is the first paper that presents a monetary policy stance index with error bands. Moreover,

we identify periods of loose and tight monetary policy, in line with previous literature, and we

also quantify the relative importance of instruments in this process. For the last decade, we

can observe on average a loose stance, a situation that is reverted before the outbreak of the

Great Recession, with a tight period governed by the drop in Non-Borrowed reserves and a

subsequent loose stance after the implementation of Unconventional Monetary Policies starting

in 2009. Furthermore, the identified monetary policy shocks exhibit an important change in

their transmission mechanism, especially for the period related with the Great Recession.

Overall, this paper presents a powerful approach that is capable of capturing different

episodes regarding monetary policy design, especially the so-called Unconventional Monetary

Policies. Even more important is the fact that we present here a monetary policy stance index

with error bands, which we expect to be useful for both policy makers and researchers.

It remains to be explored the role of communication in Unconventional Monetary Policies,

the announcement of future paths of interest rates (forward guidance) and monetary authority

reputation. We believe that these type of issues should be explored in a richer setup and we

leave it for future agenda.

28

A Sampling Parameter blocks

This section takes an extended version of the algorithm described in Canova and Perez Forero

(2015). We describe the sampling procedure for parameter blocks(BT , αT , cT ,ΣT , sT , V

)and

we do it sequentially using the logic of Gibbs Sampling (see Chib (2001)). We emphasize how

to sample blocks(αT , cT

)and we refer the reader to Primiceri (2005)’s Appendix A for specific

details regarding sampling blocks(BT ,ΣT , sT , V

).

A.1 Setting the State Space form for matrices At and C−1t

Consider the state space model generated after sampling the reduced-form coefficients Bt. From

(6) let

At

(yt −X ′tBt

)= Atyt = CtΣtεt

Then the state-space form can be written as

yt = Zα,tαt + CtΣtεt (21)

αt = αt−1 + ζt (22)

where yt and Zα,t are defined in subsection A.4, αt are the free elements in At and V ar (ζt) = Sa.

Similarly, consider the following state space model generated after sampling the vector αt.

From (6) let

C−1t At

(yt −X ′tBt

)= C−1t yt = −→y t = Σtεt

Then the state-space form can be written as

−→y t = Zc,tct + Σtεt (23)

ct = ct−1 + %t (24)

where Zc,t is also defined in subsection A.4 , ct are the free elements in C−1t and V ar (%t) = Sc.

29

A.2 The algorithm

In this section we closely follow Canova and Perez Forero (2015)’s extension to Primiceri (2005)’s

procedure. Letsl,tTt=1

Ml=1

be a discrete indicator variable which takes j = 1, . . . , k possible

values. The procedure has 8 steps and 6 sampling blocks:

1. Set an initial value for (BT0 , α

T0 , c

T0 ,Σ

T0 , s

T0 , V0) and set i = 1.

2. Draw BTi from from p

(BTi | αTi−1, cTi−1,ΣT

i−1, sTi−1, Vi−1

)· IB

(BTi

)using kalman smoothed

estimates Bt|T obtained from the system (6)− (7) and compute yTi , where IB (.) truncates

the posterior distribution to insure the stability of the companion form.

3. Draw αTi from

p(αTi | yTi , cTi−1,ΣT

i−1, sTi−1, Vi−1

)= p

(αi,T | yTi , ci−1,T ,Σi−1,T , si−1,T , Vi−1

)× (25)

T−1∏t=1

p(αi,t | αi,t+1, y

ti , ci−1,t,Σi−1,t, si−1,t, Vi−1

)∝ p

(αi,T | yTi , ci−1,T ,Σi−1,T , si−1,T , Vi−1

)×

T−1∏t=1

p(αi,t | yti , ci−1,t,Σi−1,t, si−1,t, Vi

)×

ft+1 (αi,t+1 | αi,t, ci−1,t,Σi−1,t, si−1,t, Vi−1)

4. Draw cTi from

p(cTi | −→y Ti , αTi ,ΣT

i−1, Vi−1)

= p(ci,T | −→y Ti , αi,T ,Σi−1,T , si−1,T , Vi−1

)(26)

×T−1∏t=1

p(ci,t | ci,t+1,

−→y ti, αi,t,Σi−1,t, si−1,t, Vi−1)

∝ p(ci,T | −→y Ti , αi,T ,Σi−1,T , si−1,T , Vi−1

)×T−1∏t=1

p(ci,t | −→y ti, αi,t,Σi−1,t, si−1,t, Vi−1

)×ft+1 (ci,t+1 | ci,t, αi,t,Σi−1,t, si−1,t, Vi−1)

5. Draw ΣTi using a log-normal approximation to their distribution as in Kim et al. (1998).

After sampling(BTi , α

Ti , c

Ti

), the state space is linear but the error term is not normally

30

distributed. To see this, note that given(BTi , α

Ti , c

Ti

), the state space system is

C−1t Atyt = y∗∗t = Σtεt

and (10). Consider the i− th equation y∗∗i,t = σi,tεi,t, where i = 1, . . . ,M , σi,t is the i−th

element in the main diagonal of Σt and εi,t is the i−th element of εt.

y∗t = log[(y∗∗i,t)2

+ c]≈ 2 log (σi,t) + log ε2i,t (27)

Then where c is a small constant. Since εi,t is Gaussian, log ε2i,t is log(χ2)

distributed

and we approximate this distribution by a mixture of normals. Since conditional on st,

the model is linear and Gaussian, standard Kalman Filter recursions can be used to draw

ΣtTt=1 from the system (27)− (10). To ensure independence across structural variances,

each element of the sequence σi,tMi=1 is sampled assuming that the covariance matrix W

is diagonal.

6. Draw the indicator of mixture of normals sTi . Conditional on ΣTi , y∗t , and given l and t,

we draw u ∼ U (0, 1) and compare it with the discrete distribution of sl,t which is given

by

P(sl,t = j | y∗l,t, log (σl,t)

)∝ qj × φ

(y∗l,t − 2 log (σl,t)−mj + 1.2704

υj

);

j = 1, . . . , k; l = 1, . . . ,M

where φ (.) is the probability density function of a normal distribution, and the argument

of this function is the standardized error term log ε2l,t (see Kim et al. (1998)). Then we

assign sl,t = j iff P(sl,t ≤ j − 1 | y∗l,t, log (σl,t)

)< u ≤ P

(sl,t ≤ j | y∗l,t, log (σl,t)

).

7. Draw Vi from P(Vi | αTi , cTi , yTi ,ΣT

i−1, sTi−1)

using definitions (7) − (10). The covariance

matrix Vi is sampled assuming that each block follows an independent Wishart distribu-

tion.

8. Set BTi , α

Ti , c

Ti ,Σ

Ti , s

Ti , Vi as the initial value for the next iteration and set i = i + 1.

31

Repeat 2 to 7 if i < N , otherwise stop.

The complete cycle of draws is repeated N = Nb + Nd times and the first Nb draws

are discarded to ensure convergence in distribution. Because the draws are generally serially

correlated, one every nthin of the last Nd draws is used for inference.

A.3 The details in steps 3 and 4

This subsection follows closely Canova and Perez Forero (2015). For steps 3 and 4 we use a

Metropolis step to determine whether a draw from a proposal distribution is retained or not.

We only illustrate the case of sampling vector αt, since sampling vector ct will be completely

symmetric following the Multi-move Gibbs Sampling logic. The densities p(αt | yt, ct,Σt, s, V

)are obtained applying the Extended Kalman Smoother (see subsection A.5) to the original

system (21)− (22). To draw αTi given yTi , cTi−1,Σ

Ti−1, s

Ti−1, Vi−1, we proceed as follows:

1. If i = 0, take an initial value αT0 = α0,tTt=1. If not,

2. Given yTi , cTi−1,Σ

Ti−1, s

Ti−1, Vi−1, set the state space form and compute

α∗(i)t|t+1

Tt=1

andP∗(i)t|t+1

Tt=1

using the EKS whereP ∗t|t+1

Tt=1

denotes the covariance matrix ofα∗t|t+1

Tt=1

.

3. Generate a candidate draw zT = ztTt=1, where for each t = 1, . . . , T p∗α (zt) = N(α∗(i)t|t+1, rP

∗(i)t|t+1

),

and r is a constant. Let p∗α(zT)

=T∏t=1

p∗α (zt).

4. Compute θ =p(zT )p(αT

i−1)where p (.) is the RHS of (25) using the EKS approximation. Draw

a v ∼ U (0, 1). Set αTi = zT if v < ω and set αTi = αTi−1 otherwise, where

ω ≡

min θ, 1 , if Iα(zT)

= 1

0, if Iα(zT)

= 0

and Iα (.) is a truncation indicator.

Finally, steps 2 to 4 in this sub-loop are repeated every time step 3 of the main loop is

executed.

32

A.4 The identified system

Recall the expression

Atyt = CtΣtεt

The expression of the measurement equation is possible to obtain since vec (Atyt) = vec (CtΣtεt).

Then, using the fact that At(M×M), yt(M×1), Σt(M×M), εt(M×1), then

vec (Atyt) = vec (IMAtyt) =(y′t ⊗ IM

)vec (At)

and also

vec (Ct (Σtεt)) =((Σtεt)

′ ⊗ IM)vec (Ct) = (I1 ⊗ Ct) vec (Σtεt) = CtΣtεt

since Atyt and CtΣtεt are already column vectors18. On the other hand, following Amisano

and Giannini (1997) and Hamilton (1994), we also know that the matrix of the SVAR can be

decomposed as follows

vec (At) = SAαt + sA (28)

vec(C−1t

)= SCF (ct) + sC (29)

where SA(M2×dimα), sA(M2×1), SC(M2×dimF (c)) and sC(M2×1) are matrices filled by ones and

zeros. Moreover, αt, ct and F (ct) are the vectors of free parameters in At, Ct and C−1t ,

respectively and F (.) : Rdim(c) → RdimF (c) is in general a nonlinear invertible function. That

is, we sample the vector F (ct)Tt=1 and if and only if F (.) is invertible, then we can recover

ctTt=1 =F−1 [F (ct)]

Tt=1

. We will denote ct = F (ct). Collecting all the results we get

(y′t ⊗ IM

)(SAαt + sA) = CtΣtεt

18In general, we have applied the property ABd = (d′ ⊗A) vec (B), where A is an m×n matrix, B is an n× qmatrix and d is a (q × 1) vector. See details in Magnus and Neudecker (2007), chapter 2, pp. 35.

33

Rewriting this equation

(y′t ⊗ IM

)SAαt +

(y′t ⊗ IM

)sA = CtΣtεt

(y′t ⊗ IM

)sA = −

(y′t ⊗ IM

)SAαt + CtΣtεt

The state space form is now

yt = Zα,tαt + CtΣtεt

αt = αt−1 + ζt

where

yt ≡(y′t ⊗ IM

)sA (30)

Zα,t ≡ −(y′t ⊗ IM

)SA (31)

On the other hand, given αt,we proceed in the following way

vec(C−1t Atyt

)= vec ((Σtεt)) = Σtεt

((Atyt)

′ ⊗ IM)vec

(C−1t

)= Σtεt

((Atyt)

′ ⊗ IM)

(SC ct + sC) = Σtεt

((Atyt)

′ ⊗ IM)sC = −

((Atyt)

′ ⊗ IM)SC ct + Σtεt

The state space form is now

−→y t = ZC,tct + Σtεt

ct = ct−1 + %t

where

−→y t ≡((Atyt)

′ ⊗ IM)sC (32)

ZC,t ≡ −((Atyt)

′ ⊗ IM)SC (33)

34

Moreover, for the specific case of the model presented, we have that yt denotes the residuals for

the first stage and also the matrices

At =

1 0 0 0 0 0

απx,t 1 0 0 0 0

αcx,t αcπ,t 1 0 0 0

αTRx,t αTRπ,t αTRc,t 1 αd1,t 0

αFFRx,t αFFRπ,t αFFRc,t −1/αb1,t 1 1/αb1,t

αNBRx,t αNBRπ,t αNBRc,t 0 0 1

(34)

and

Ct =

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 φdt φbt 1

(35)

therefore, M = 6, dimα = 13. As it is evident, Ct is a lower-triangular matrix with the main

diagonal governed by ones, i.e. a unitriangular matrix. Moreover, C ′t will be unitriangular as

well and in this case it can also be classified as a Frobenius matrix19. The inverse of a Frobenius

matrix X is exactly X−1 = −X. Thus, provided by the fact that [C ′t]−1 =

[C−1t

]′, we have that

[C ′t]−1 =

[C−1t

]′= −C ′t ⇒ C−1t = −Ct, hence

C−1t =

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 −φdt −φbt 1

19http://en.wikipedia.org/wiki/Frobenius matrix

35

and dimF (c) = dim c = 2. Therefore, it turns out that in this particular case the function F (.)

is actually a linear transformation, i.e. ct = F (ct) = −ct. That is, as long as C ′t is Frobenius

matrix, F (.) will be linear with a well defined F−1 (.).

As a result, we need to define matrices SA, sA, SC , sC filled by 0s and 1s. These matrices,

together with the column vectors

αt =

απx,t, αPcomx,t , αTRx,t , α

NBRx,t , αFFRx,t , αPcπ,t, α

TRπ,t ,

αNBRπ,t , αFFRπ,t , αTRPc,t, αNBRPc,t , α

FFRPc,t , α

b1,t

′

and

ct =[−φdt ,−φbt

]′are set such that equations (28) and (29) hold exactly.

A.5 Extended Kalman Smoother

This subsection follows closely Canova and Perez Forero (2015). Consider the nonlinear system

yt = zt (αt, εt)

αt = tt (αt−1, ηt)

with V ar (εt) = Qεt and V ar (ηt) = Qηt . To apply the EKS to our system of equations, we first

compute the jacobians:

Zα,t =∂zt∂αt

(at|t−1, 0

); Zε,t =

∂zt∂εt

(at|t−1, 0

)(36)

Tα,t =∂tt∂αt

(at|t−1, 0

); Tη,t =

∂tt∂ηt

(at|t−1, 0

)and we predict the mean and variance at each t = 1, . . . , T :

at|t−1 = tt−1(at−1|t−1, 0

)

36

Pt|t−1 = Tα,tPt−1|t−1T′α,t + Tη,tQ

ηt T′η,t

The Kalman gain is computed:

Ωt = Zα,tPt|t−1Z′α,t + Zε,tQ

εt Z′ε,t

Kt = Pt|t−1Z′α,tΩ

−1t

As new information arrives, estimates of αt and variance are updated according to

at|t = at|t−1 +Kt

[yt − zt

(at|t−1, 0

)]Pt|t = Pt|t−1 − Pt|t−1Z ′α,tΩ−1t Zα,tP

′t|t−1

To smooth the estimates, set α∗T |T = aT |T , P ∗T |T = PT |T and , for t = T − 1, . . . , 1, compute

α∗t|t+1 = at|t + Pt|tT′α,tP

−1t+1|t

(α∗t+1|t+2 − tt−1

(at|t, 0

))

P ∗t|t+1 = Pt|t − Pt|tT ′α,tP−1t+1|tTα,tP′t|t−1

To start the iterations we use fixed values a1|0 = αN×1 and P0|0 = IN .

Finally, notice that the original tt (.) and zt (.) are used for computing prediction and up-

dating equations of αt.

B Diagnosis of convergence of the Markov Chain to the Ergodic

Distribution

Following Geweke (1992), Primiceri (2005) and Baumeister and Benati (2013) among others,

we check for the autocorrelation properties of the different blocks of the Markov Chain via the

inefficiency factor. Let the Relative Numerical Efficiency (RNE) be:

RNE =1

2π

1

S (0)

∫ π

−πS (ω) dω (37)

37

where S (.) is the spectral density of an element of the Markov Chain. The inefficiency factor

IF = 1/RNE, can be interpreted as an estimate of (1 + 2∑∞

k=1 ρk), where ρk denotes the

autocorrelation of k-th order. Thus, high values of IF indicate strong serial correlation across

draws. We use the MATLAB file coda.m from James P. Lesage toolbox to calculate the IF. We

set a 4% tapered window for the estimation of the spectral density at frequency zero and take

values around or below 20 as cut-off point (and values below are considered as satisfactory).

Figure 8 depicts the IF value for each parameter of the model. Overall, serial correlation across

draws does not seem to be an issue.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

x 104

0

0.5

1

1.5

2

2.5

3

3.5

Figure 8: Inefficiency Factor IF for each parameter in the model

38

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